Sparse polynomial interpolation without and with errors

We present algorithms for performing sparse univariate
polynomial interpolation with errors in the evaluations of
the polynomial. Our interpolation algorithms use as a
substep an algorithm that originally is by R. Prony from
the French Revolution (Year III, 1795) for interpolating
exponential sums and which is rediscovered to decode
digital error correcting BCH codes over finite fields (1960).
Since Prony's algorithm is quite simple, we will give
a complete description, as an alternative for Lagrange/Newton
interpolation for sparse polynomials. When very few errors
in the evaluations are permitted, multiple sparse interpolants
are possible over finite fields or the complex numbers,
but not over the real numbers. The problem is then a simple
example of list-decoding in the sense of Guruswami-Sudan.
Finally, we present a connection to the Erdoes-Turan Conjecture
(Szemeredi's Theorem).
This is joint work with Clement Pernet, Univ. Grenoble.